422:
in the nineteenth century. The more strict analogy expressed by the 'global field' idea, in which a
Riemann surface's aspect as algebraic curve is mapped to curves defined over a finite field, was built up during the 1930s, culminating in the
692:
1075:
Algebraic number theory. Proceedings of an instructional conference organized by the London
Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union
390:
129:
71:
166:
578:
915:
876:
424:
438:
It is usually easier to work in the function field case and then try to develop parallel techniques on the number field side. The development of
881:
459:
301:
There are a number of formal similarities between the two kinds of fields. A field of either type has the property that all of its
285:
consists of such local data that agree on the intersections of open affines. This technically defines the rational functions on
193:
79:
1103:
997:
265:
is the set of all rational functions on that variety. On an irreducible algebraic curve (i.e. a one-dimensional variety
1045:
974:
455:
1127:
989:
1122:
339:
486:
480:
96:
48:
966:
1132:
502:
54:
407:
734:
189:
44:
687:{\displaystyle \theta _{v}:K_{v}^{\times }/N_{L_{v}/K_{v}}(L_{v}^{\times })\to G^{\text{ab}},}
514:
306:
138:
1007:
941:
904:
466:
also made use of techniques that reduced the number field case to the function field case.
293:
of the affine coordinate ring of any open affine subset, since all such subsets are dense.
1082:
1070:
1055:
1015:
8:
520:
400:
322:
223:
215:
173:
37:
33:
25:
447:
314:
290:
981:
879:(1945), "Axiomatic characterization of fields by the product formula for valuations",
1099:
1089:
1062:
1041:
1023:
993:
970:
463:
419:
262:
132:
83:
932:
895:
762:
by multiplying the local components of an idĂšle class. One of the statements of the
1078:
1066:
1051:
1033:
1011:
927:
890:
721:
557:
415:
406:
The analogy between the two kinds of fields has been a strong motivating force in
1093:
1037:
1003:
937:
900:
532:
439:
411:
318:
242:
227:
219:
86:
269:) over a finite field, we say that a rational function on an open affine subset
959:
947:
494:
451:
450:
is a dramatic example. The analogy was also influential in the development of
302:
274:
1116:
490:
443:
428:
1028:
524:
246:
90:
951:
561:
310:
29:
17:
257:
The function field of an irreducible algebraic curve over a finite field
911:
872:
177:
497:
over a global field are equivalent if and only if they are equivalent
848:
296:
1032:, Graduate Texts in Mathematics, vol. 67, translated by
988:. Vol. 322. Translated by Schappacher, Norbert. Berlin:
776:
431:
in 1940. The terminology may be due to Weil, who wrote his
918:(1946), "A note on axiomatic characterization of fields",
535:. It can be described in terms of cohomology as follows:
800:
788:
836:
313:). Every field of either type can be realized as the
581:
519:
Artin's reciprocity law implies a description of the
342:
141:
99:
57:
824:
812:
410:. The idea of an analogy between number fields and
686:
384:
273:is defined as the ratio of two polynomials in the
172:An axiomatic characterization of these fields via
160:
123:
65:
766:is that this results in a canonical isomorphism.
1114:
425:Riemann hypothesis for curves over finite fields
325:is of finite index. In each case, one has the
1077:, London: Academic Press, pp. 128â161,
910:
871:
806:
794:
435:(1967) in part to work out the parallelism.
1065:(1967), "VI. Local class field theory", in
297:Analogies between the two classes of fields
135:in one variable over the finite field with
474:
1098:, Springer Science & Business Media,
950:, "Global fields", in J.W.S. Cassels and
931:
894:
281:, and that a rational function on all of
102:
59:
980:
854:
842:
782:
508:
1115:
194:Function field of an algebraic variety
93:, equivalently, a finite extension of
1088:
1061:
1022:
830:
818:
183:
385:{\displaystyle \prod _{v}|x|_{v}=1,}
124:{\displaystyle \mathbb {F} _{q}(T)}
13:
572:describes a canonical isomorphism
14:
1144:
962:, 1973. Chap.II, pp. 45â84.
180:and George Whaples in the 1940s.
1036:, New York, Heidelberg, Berlin:
965:J.W.S. Cassels, "Local fields",
36:. There are two kinds of global
933:10.1090/S0002-9904-1946-08549-3
896:10.1090/S0002-9904-1945-08383-9
758:can be assembled into a single
32:) that are characterized using
668:
665:
647:
363:
354:
118:
112:
1:
864:
501:, i.e. equivalent over every
769:
533:Hasse localâglobal principle
66:{\displaystyle \mathbb {Q} }
7:
857:, p. 300, Theorem 6.3.
489:is a fundamental result in
469:
10:
1149:
967:Cambridge University Press
512:
478:
210:An algebraic number field
187:
237:is a field that contains
206:An algebraic number field
202:is one of the following:
807:Artin & Whaples 1946
795:Artin & Whaples 1945
442:and its exploitation by
321:in which every non-zero
1128:Algebraic number theory
986:Algebraic Number Theory
956:Algebraic number theory
487:HasseâMinkowski theorem
481:HasseâMinkowski theorem
475:HasseâMinkowski theorem
408:algebraic number theory
261:A function field of an
214:is a finite (and hence
161:{\displaystyle q=p^{n}}
24:is one of two types of
920:Bull. Amer. Math. Soc.
882:Bull. Amer. Math. Soc.
785:, p. 134, Sec. 5.
688:
386:
329:for non-zero elements
307:locally compact fields
275:affine coordinate ring
190:Algebraic number field
162:
125:
67:
45:Algebraic number field
1034:Greenberg, Marvin Jay
764:Artin reciprocity law
750:for different places
724:of global fields and
703:local reciprocity map
689:
570:local reciprocity law
531:that is based on the
515:Artin reciprocity law
509:Artin reciprocity law
499:locally at all places
493:that states that two
387:
245:when considered as a
163:
126:
76:Global function field
68:
579:
446:in his proof of the
340:
139:
97:
55:
1123:Field (mathematics)
707:norm residue symbol
664:
609:
458:. The proof of the
433:Basic Number Theory
1090:Serre, Jean-Pierre
1063:Serre, Jean-Pierre
1024:Serre, Jean-Pierre
699:local Artin symbol
684:
650:
595:
564:with Galois group
527:of a global field
448:Mordell conjecture
382:
352:
315:field of fractions
291:field of fractions
184:Formal definitions
158:
133:rational functions
121:
63:
28:(the other one is
1105:978-1-4757-5673-9
999:978-3-540-65399-8
760:global symbol map
735:idĂšle class group
678:
464:Langlands program
460:fundamental lemma
420:Heinrich M. Weber
343:
263:algebraic variety
1140:
1133:Algebraic curves
1108:
1092:(29 June 2013),
1085:
1058:
1019:
982:Neukirch, JĂŒrgen
944:
935:
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858:
852:
846:
840:
834:
828:
822:
816:
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786:
780:
722:Galois extension
693:
691:
690:
685:
680:
679:
676:
663:
658:
646:
645:
644:
643:
634:
629:
628:
614:
608:
603:
591:
590:
558:Galois extension
523:of the absolute
416:Richard Dedekind
412:Riemann surfaces
399:varies over all
391:
389:
388:
383:
372:
371:
366:
357:
351:
228:rational numbers
174:valuation theory
167:
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62:
49:finite extension
1148:
1147:
1143:
1142:
1141:
1139:
1138:
1137:
1113:
1112:
1111:
1106:
1067:Cassels, J.W.S.
1048:
1038:Springer-Verlag
1000:
990:Springer-Verlag
916:Whaples, George
877:Whaples, George
867:
862:
861:
853:
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841:
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580:
577:
576:
555:
546:
517:
511:
495:quadratic forms
483:
477:
472:
456:Main Conjecture
440:Arakelov theory
403:of the field.
367:
362:
361:
353:
347:
341:
338:
337:
327:product formula
319:Dedekind domain
299:
241:and has finite
220:field extension
196:
188:Main articles:
186:
152:
148:
140:
137:
136:
131:, the field of
106:
101:
100:
98:
95:
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87:algebraic curve
58:
56:
53:
52:
12:
11:
5:
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1059:
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998:
978:
963:
960:Academic Press
948:J.W.S. Cassels
945:
926:(4): 245â247,
908:
889:(7): 469â492,
868:
866:
863:
860:
859:
847:
845:, p. 391.
835:
833:, p. 197.
823:
821:, p. 140.
811:
799:
787:
774:
773:
771:
768:
745:
733:stand for the
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551:
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521:abelianization
513:Main article:
510:
507:
505:of the field.
479:Main article:
476:
473:
471:
468:
452:Iwasawa theory
393:
392:
381:
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80:function field
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61:
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2:
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1047:3-540-90424-7
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1005:
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987:
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975:0-521-31525-5
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968:
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869:
856:
855:Neukirch 1999
851:
844:
843:Neukirch 1999
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783:Neukirch 1999
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491:number theory
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482:
467:
465:
461:
457:
453:
449:
445:
444:Gerd Faltings
441:
436:
434:
430:
426:
421:
417:
414:goes back to
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181:
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176:was given by
175:
153:
149:
145:
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134:
115:
107:
92:
88:
85:
81:
77:
74:
50:
46:
43:
42:
41:
39:
35:
31:
27:
23:
19:
1095:Local Fields
1094:
1074:
1071:Fröhlich, A.
1029:Local Fields
1027:
985:
955:
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706:
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698:
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569:
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562:local fields
552:
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539:
537:
528:
525:Galois group
518:
498:
484:
437:
432:
405:
396:
394:
330:
326:
311:local fields
300:
286:
282:
278:
270:
266:
260:
250:
247:vector space
238:
234:
230:
211:
209:
200:global field
199:
197:
171:
91:finite field
75:
30:local fields
22:global field
21:
15:
952:A. Frohlich
912:Artin, Emil
873:Artin, Emil
741:. The maps
697:called the
427:settled by
303:completions
84:irreducible
18:mathematics
1117:Categories
1083:0153.07403
1056:0423.12016
1016:0956.11021
865:References
831:Serre 1979
819:Serre 1967
503:completion
429:André Weil
401:valuations
289:to be the
178:Emil Artin
34:valuations
770:Citations
669:→
661:×
606:×
584:θ
345:∏
243:dimension
216:algebraic
168:elements.
1073:(eds.),
1026:(1979),
984:(1999).
977:. P.56.
969:, 1986,
470:Theorems
454:and the
233:. Thus
1008:1697859
954:(eds),
942:0015382
905:0013145
705:or the
462:in the
222:of the
89:over a
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940:
903:
743:θ
701:, the
568:. The
395:where
82:of an
78:: The
38:fields
26:fields
720:be a
556:be a
323:ideal
317:of a
309:(see
249:over
224:field
1100:ISBN
1042:ISBN
994:ISBN
971:ISBN
712:Let
538:Let
485:The
418:and
305:are
192:and
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1079:Zbl
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737:of
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1002:.
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